Some applications of the generalized Eulerian numbers

Rza̧dkowski, Grzegorz; Urlińska, Małgorzata

doi:10.1016/j.jcta.2018.11.012

Cited by 8 publications

(6 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Section: Introductionmentioning

confidence: 99%

“…In the past decades, there has been much work on Eulerian polynomial and its generalizations (see [18,21,30,37] for instance). For example, by using a kind of first-order differential equation, Rzadkowski and Urlińska [30] considered a unified generalization of Eulerian polynomials and second-order Eulerian polynomials. In the following we first recall the definitions of k-order Eulerian polynomials and 1/k-Eulerian polynomials, and then we present summation formulas for these polynomials.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux

Han¹,

Ma²

2020

Preprint

View full text Add to dashboard Cite

In this paper we first present summation formulas for k-order Eulerian polynomials and 1/k-Eulerian polynomials. We then present combinatorial expansions of (c(x)D) n in terms of inversion sequences as well as k-Young tableaux, where c(x) is a differentiable function in the indeterminate x and D is the derivative with respect to x. We define the g-indexes of k-Young tableaux and Young tableaux, which have important applications in combinatorics.By establishing some relations between k-Young tableaux and standard Young tableaux, we express Eulerian polynomials, second-order Eulerian polynomials, André polynomials and the generating polynomials of gamma coefficients of Eulerian polynomials in terms of standard Young tableaux, which imply a deep connection among these polynomials.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux

Han¹,

Ma²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In our further work, we intend to use, in a similar way, the logistic wavelets of higher order (see Section 2.3). Moreover, using appropriate special numbers we are going to define analogous wavelets for the Gompertz function (see some initial calculations [29,30]) or for some generalizations of the logistic function (for preliminary theorems see [31]).…”

Section: Discussionmentioning

confidence: 99%

Logistic Wavelets and Their Application to Model the Spread of COVID-19 Pandemic

Rza̧dkowski

Figlia²

2021

Applied Sciences

Self Cite

View full text Add to dashboard Cite

In the present paper, we model the cumulative number of persons, reported to be infected with COVID-19 virus, by a sum of several logistic functions (the so-called multilogistic function). We introduce logistic wavelets and describe their properties in terms of Eulerian numbers. Moreover, we implement the logistic wavelets into Matlab’s Wavelet Toolbox and then we use the continuous wavelet transform (CWT) to estimate the parameters of the approximating multilogistic function. Using the examples of several countries, we show that this method is effective as a method of fitting a curve to existing data. However, it also has a predictive value, and, in particular, allows for an early assessment of the size of the emerging new wave of the epidemic, thus it can be used as an early warning method.

show abstract

“…In addition, different generalizations of Eulerian polynomials and Eulerian numbers were considered, see Haglund-Zhang [24], Han-Mao-Zeng [25], Rzadkowski-Urlińska [35], Zhu [48], Zhuang [52]. But recently in many new combinatorial enumerations, there bring out more and more combinatorial triangles satisfying certain three-term recurrence similar to recurrence relations (1.8), (1.9) and (1.11).…”

Section: Structure Of This Papermentioning

confidence: 99%

A unified approach to combinatorial triangles: a generalized Eulerian polynomial

Zhu

2020

Preprint

View full text Add to dashboard Cite

Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array [T n,k ] n,k≥0 satisfying the recurrence relation:We derive a functional transformation for its row-generating function T n (x) from the row-generating function A n (x) of another array [A n,k ] n,k satisfying a two-term recurrence relation. Based on this transformation, we can get properties of T n,k and T n (x) including nonnegativity, log-concavity, real rootedness, explicit formula and so on. Then we extend the famous Frobenius formula, the γ positivity decomposition and the David-Barton formula for the classical Eulerian polynomial to those of a generalized Eulerian polynomial. We also get an identity for the generalized Eulerian polynomial with the general derivative polynomial. Finally, we apply our results to an array from the Lambert function, a triangular array from staircase tableaux and the alternating-runs triangle of type B in a unified approach.

show abstract

Some applications of the generalized Eulerian numbers

Cited by 8 publications

References 11 publications

Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux

Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux

Logistic Wavelets and Their Application to Model the Spread of COVID-19 Pandemic

A unified approach to combinatorial triangles: a generalized Eulerian polynomial

Contact Info

Product

Resources

About